A computational procedure for analysis of fractures in two-dimensional multi-field media

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Abstract

A systematic procedure is followed to develop singularity-reduced integral equations for modeling cracks in two-dimensional, linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations (IEs) for "generalized displacement" and "generalized stress"; these serve as the basis for the development of a Symmetric Galerkin Boundary Element Method (SGBEM). The implementation is carried out to allow treatment of general mixed boundary conditions, an arbitrary number of cracks, and multi-region domains (in which regions having different material properties are bonded together). Finally, a procedure for calculation of T-stress, the constant term in the asymptotic series expansion of crack-tip stress field, is developed for anisotropic elastic media.
The pair of weak-form boundary IEs that is derived (one for generalized displacement and the other one for generalized stress) are completely regularized in the sense that all kernels that appear are (at most) weakly-singular. This feature allows standard Co elements to be utilized in the SGBEM, and such elements are employed everywhere except at the crack tip. A special crack-tip element is developed to properly model the asymptotic behavior of the relative crack-face displacements. This special element contains "extra" degrees of freedom that allow the generalized stress intensity factors to be directly obtained from the solution of the governing system of discretized equations. It should be noted that while the integral equations contain only weakly-singular kernels (and so are integrable in the usual sense) there remains a need to devise special integration techniques to accurately evaluate these integrals as part of the numerical implementation.
Various examples for crack problems are treated to illustrate the accuracy and versatility of the proposed procedure for both unbounded and finite domains and for both single-region and multi-region problems. It is found that highly accurate fracture data can be obtained using relatively course meshes.
Finally, this dissertation addresses the development of a numerical procedure to calculate T-stress for crack problems in general anisotropic elastic media. T-stress is obtained from the sum of crack-face displacements which are computed via a (regularized) integral equation of the boundary data. Two approaches for computing the derivative of the sum of crack-face displacements are proposed: one uses numerical differentiation, and the other one uses a weak-form integral equation. Various examples are examined to demonstrate that highly accurate results are obtained by means of both approaches.